Editing Norm (mathematics) (section)

==Definition==

Given a [[vector space]] <math>X</math> over a [[Field extension|subfield]] <math>F</math> of the  complex numbers <math>\Complex,</math> a '''norm''' on <math>X</math> is a [[real-valued function]] <math>p : X \to \Reals</math> with the following properties, where <math>|s|</math> denotes the usual [[absolute value]] of a scalar <math>s</math>:<ref>{{cite book|title=Real Mathematical Analysis |publisher=Springer |author=Pugh, C.C.|year=2015|page=[https://books.google.com/books?id=2NVJCgAAQBAJ&pg=PA28 page 28]|isbn=978-3-319-17770-0}} {{cite book|title=Quantum Mechanics in Hilbert Space|author=Prugovečki, E.|year=1981|page=[https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA20 page 20]}}</ref>

# [[Subadditive function|Subadditivity]]/[[Triangle inequality]]: <math>p(x + y) \leq p(x) + p(y)</math> for all <math>x, y \in X.</math>
# [[Homogeneous function|Absolute homogeneity]]: <math>p(s x) = |s| p(x)</math> for all <math>x \in X</math> and all scalars <math>s.</math>
# [[Positive definiteness]]/Positiveness{{sfn|Kubrusly|2011|p=200}}/{{Visible anchor|Point-separating}}: for all <math>x \in X,</math> if <math>p(x) = 0</math> then <math>x = 0.</math>
#* Because property (2.) implies <math>p(0) = 0,</math> some authors replace property (3.) with the equivalent condition: for every <math>x \in X,</math> <math>p(x) = 0</math> if and only if <math>x = 0.</math>

A [[seminorm]] on <math>X</math> is a function <math>p : X \to \Reals</math> that has properties (1.) and (2.)<ref>{{cite book|title=Functional Analysis|author=Rudin, W.|year=1991|page=25}}</ref> so that in particular, every norm is also a seminorm (and thus also a [[sublinear function]]al). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if <math>p</math> is a norm (or more generally, a seminorm) then <math>p(0) = 0</math> and that <math>p</math> also has the following property:

#<li value="4">[[Nonnegative|Non-negativity]]:{{sfn|Kubrusly|2011|p=200}} <math>p(x) \geq 0</math> for all <math>x \in X.</math></li>

Some authors include non-negativity as part of the definition of "norm", although this is not necessary.
Although this article defined "{{em|positive}}" to be a synonym of "positive definite", some authors instead define "{{em|positive}}" to be a synonym of "non-negative";{{sfn|Narici|Beckenstein|2011|pp=120-121}} these definitions are not equivalent.

===Equivalent norms===

Suppose that <math>p</math> and <math>q</math> are two norms (or seminorms) on a vector space <math>X.</math> Then <math>p</math> and <math>q</math> are called '''equivalent''', if there exist two positive real constants <math>c</math> and <math>C</math> such that for every vector <math>x \in X,</math>
<math display="block">c q(x) \leq p(x) \leq C q(x).</math>
The relation "<math>p</math> is equivalent to <math>q</math>" is [[Reflexive relation|reflexive]], [[Symmetric relation|symmetric]] (<math>c q \leq p \leq C q</math> implies <math>\tfrac{1}{C} p \leq q \leq \tfrac{1}{c} p</math>), and [[Transitive relation|transitive]] and thus defines an [[equivalence relation]] on the set of all norms on <math>X.</math>
The norms <math>p</math> and <math>q</math> are equivalent if and only if they induce the same topology on <math>X.</math><ref name="Conrad Equiv norms">{{cite web |url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/equivnorms.pdf |title=Equivalence of norms |last=Conrad |first=Keith |website=kconrad.math.uconn.edu |access-date=September 7, 2020 }}</ref> Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.<ref name="Conrad Equiv norms"/>

===Notation===

If a norm <math>p : X \to \R</math> is given on a vector space <math>X,</math> then the norm of a vector <math>z \in X</math> is usually denoted by enclosing it within double vertical lines: <math>\|z\| = p(z)</math>, as proposed by [[Stefan Banach]] in his doctoral thesis from 1920.  Such notation is also sometimes used if <math>p</math> is only a seminorm.  For the length of a vector in Euclidean space (which is an example of a norm, as [[#Euclidean norm|explained below]]), the notation <math>|x|</math> with single vertical lines is also widespread.